Lcm of degrees of irreducible representations
This article defines an arithmetic function on groups
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This term is related to: linear representation theory
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Contents
Definition
For a group over a field
Suppose is a group and
is a field. The lcm of degrees of irreducible representations of
is defined as the least common multiple of all the degrees of irreducible representations of
over
.
Typical context: finite group and splitting field
The typical context is where is a finite group and
is a splitting field for
. In particular, the characteristic of
is either zero or is a prime not dividing the order of
, and every irreducible representation of
over any extension field of
can be realized over
.
Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field . This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.
Default case: characteristic zero
By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take .
Related facts
What it divides
Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below:
- Degree of irreducible representation divides order of group: Hence, the lcm of degrees of irreducible representations divides the order of the whole group.
- Degree of irreducible representation divides index of center: Hence, the lcm of degrees of irreducible representations divides the index of the center, which is also the order of the inner automorphism group.
- Degree of irreducible representation divides index of abelian normal subgroup: Hence, the lcm of degrees of irreducible representations divides the index of any abelian normal subgroup.
Subgroups, quotients, direct products
- lcm of degrees of irreducible representations of subgroup divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of quotient group divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of direct product is lcm of lcms of degrees of irreducible representations of each direct factor